Moderate -0.3 This is a straightforward separable variables question requiring separation, integration of standard forms (including a simple substitution for the denominator), and application of initial conditions. The integration is routine for A-level (ln of a linear function of e^(3x)), making it slightly easier than average but still requiring proper technique.
4 The variables \(x\) and \(y\) are related by the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 y \mathrm { e } ^ { 3 x } } { 2 + \mathrm { e } ^ { 3 x } }$$
Given that \(y = 36\) when \(x = 0\), find an expression for \(y\) in terms of \(x\).
Separate variables correctly and recognisable attempt at integration of at least one side
M1
Obtain \(\ln y\), or equivalent
B1
Obtain \(k\ln(2 + e^{3x})\)
B1
Use \(y(0) = 36\) to find constant in \(y = A(2 + e^x)^k\) or \(\ln y = k\ln(2 + e^{3x}) + c\) or equivalent
M1*
Obtain equation correctly without logarithms from \(\ln y = \ln\left[A(2 + e^{3x})^k\right]\)
*M1
Obtain \(y = 4(2 + e^x)^2\)
A1
[6]
Separate variables correctly and recognisable attempt at integration of at least one side | M1 |
Obtain $\ln y$, or equivalent | B1 |
Obtain $k\ln(2 + e^{3x})$ | B1 |
Use $y(0) = 36$ to find constant in $y = A(2 + e^x)^k$ or $\ln y = k\ln(2 + e^{3x}) + c$ or equivalent | M1* |
Obtain equation correctly without logarithms from $\ln y = \ln\left[A(2 + e^{3x})^k\right]$ | *M1 |
Obtain $y = 4(2 + e^x)^2$ | A1 | [6]
4 The variables $x$ and $y$ are related by the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 y \mathrm { e } ^ { 3 x } } { 2 + \mathrm { e } ^ { 3 x } }$$
Given that $y = 36$ when $x = 0$, find an expression for $y$ in terms of $x$.
\hfill \mbox{\textit{CAIE P3 2014 Q4 [6]}}